Proof that a r.v. with a particular random walk is a martingale
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Let $Ain N+$. Let $(X_n)_{nge 1}$ be a sequence of i.i.d. random variables such that
$P(X_1=1)=1-P(X_1=-1)=p$
$p in (0, frac{1}{2})$
Consider a random walk $S_n=sum_{i=0}^n X_i$
Let $mathcal{F}_n=sigma(X_1,....,X_n)$ and define the stopping time $tau$ such that
$tau = min [n in mathbb{N}:S_n=0]$
Define
$W_n=(S_n -A-(2p-1)n)^2 - n(1-(2p-1)^2)$
I need to show that $(W_n,mathcal{F}_n)$ is a martingale
Proof until now:
I have tried to show this starting from $n=0$ and $n=1$ and eventually going true a proof through induction but, when I substitute the values in $n$, I find out that $E(W_0)=A^2$ while $E(W_1)=A^2-1+(2p-1)^2$ since I considered $E(S_1)=(2p-1)$ and it gets deleted by the same negative term in the first bracket.
probability martingales random-walk
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up vote
2
down vote
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Let $Ain N+$. Let $(X_n)_{nge 1}$ be a sequence of i.i.d. random variables such that
$P(X_1=1)=1-P(X_1=-1)=p$
$p in (0, frac{1}{2})$
Consider a random walk $S_n=sum_{i=0}^n X_i$
Let $mathcal{F}_n=sigma(X_1,....,X_n)$ and define the stopping time $tau$ such that
$tau = min [n in mathbb{N}:S_n=0]$
Define
$W_n=(S_n -A-(2p-1)n)^2 - n(1-(2p-1)^2)$
I need to show that $(W_n,mathcal{F}_n)$ is a martingale
Proof until now:
I have tried to show this starting from $n=0$ and $n=1$ and eventually going true a proof through induction but, when I substitute the values in $n$, I find out that $E(W_0)=A^2$ while $E(W_1)=A^2-1+(2p-1)^2$ since I considered $E(S_1)=(2p-1)$ and it gets deleted by the same negative term in the first bracket.
probability martingales random-walk
New contributor
$W_n$ doesn't seem to involve $tau$. What is the role of $tau$ in this question?
– Kavi Rama Murthy
Nov 22 at 10:11
Your computation of $EW_0$ and $EW_1$ shows that ${W_n}$ cannot be a martingale unless $p$ is $0$ or $1$.
– Kavi Rama Murthy
Nov 22 at 10:14
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let $Ain N+$. Let $(X_n)_{nge 1}$ be a sequence of i.i.d. random variables such that
$P(X_1=1)=1-P(X_1=-1)=p$
$p in (0, frac{1}{2})$
Consider a random walk $S_n=sum_{i=0}^n X_i$
Let $mathcal{F}_n=sigma(X_1,....,X_n)$ and define the stopping time $tau$ such that
$tau = min [n in mathbb{N}:S_n=0]$
Define
$W_n=(S_n -A-(2p-1)n)^2 - n(1-(2p-1)^2)$
I need to show that $(W_n,mathcal{F}_n)$ is a martingale
Proof until now:
I have tried to show this starting from $n=0$ and $n=1$ and eventually going true a proof through induction but, when I substitute the values in $n$, I find out that $E(W_0)=A^2$ while $E(W_1)=A^2-1+(2p-1)^2$ since I considered $E(S_1)=(2p-1)$ and it gets deleted by the same negative term in the first bracket.
probability martingales random-walk
New contributor
Let $Ain N+$. Let $(X_n)_{nge 1}$ be a sequence of i.i.d. random variables such that
$P(X_1=1)=1-P(X_1=-1)=p$
$p in (0, frac{1}{2})$
Consider a random walk $S_n=sum_{i=0}^n X_i$
Let $mathcal{F}_n=sigma(X_1,....,X_n)$ and define the stopping time $tau$ such that
$tau = min [n in mathbb{N}:S_n=0]$
Define
$W_n=(S_n -A-(2p-1)n)^2 - n(1-(2p-1)^2)$
I need to show that $(W_n,mathcal{F}_n)$ is a martingale
Proof until now:
I have tried to show this starting from $n=0$ and $n=1$ and eventually going true a proof through induction but, when I substitute the values in $n$, I find out that $E(W_0)=A^2$ while $E(W_1)=A^2-1+(2p-1)^2$ since I considered $E(S_1)=(2p-1)$ and it gets deleted by the same negative term in the first bracket.
probability martingales random-walk
probability martingales random-walk
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New contributor
edited Nov 22 at 10:50
Bernard
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116k637108
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asked Nov 22 at 10:08
user618532
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$W_n$ doesn't seem to involve $tau$. What is the role of $tau$ in this question?
– Kavi Rama Murthy
Nov 22 at 10:11
Your computation of $EW_0$ and $EW_1$ shows that ${W_n}$ cannot be a martingale unless $p$ is $0$ or $1$.
– Kavi Rama Murthy
Nov 22 at 10:14
add a comment |
$W_n$ doesn't seem to involve $tau$. What is the role of $tau$ in this question?
– Kavi Rama Murthy
Nov 22 at 10:11
Your computation of $EW_0$ and $EW_1$ shows that ${W_n}$ cannot be a martingale unless $p$ is $0$ or $1$.
– Kavi Rama Murthy
Nov 22 at 10:14
$W_n$ doesn't seem to involve $tau$. What is the role of $tau$ in this question?
– Kavi Rama Murthy
Nov 22 at 10:11
$W_n$ doesn't seem to involve $tau$. What is the role of $tau$ in this question?
– Kavi Rama Murthy
Nov 22 at 10:11
Your computation of $EW_0$ and $EW_1$ shows that ${W_n}$ cannot be a martingale unless $p$ is $0$ or $1$.
– Kavi Rama Murthy
Nov 22 at 10:14
Your computation of $EW_0$ and $EW_1$ shows that ${W_n}$ cannot be a martingale unless $p$ is $0$ or $1$.
– Kavi Rama Murthy
Nov 22 at 10:14
add a comment |
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$W_n$ doesn't seem to involve $tau$. What is the role of $tau$ in this question?
– Kavi Rama Murthy
Nov 22 at 10:11
Your computation of $EW_0$ and $EW_1$ shows that ${W_n}$ cannot be a martingale unless $p$ is $0$ or $1$.
– Kavi Rama Murthy
Nov 22 at 10:14